Chapter 7.7, Problem 22P

(A)

Interpretation Introduction

Interpretation:

The rate at which work is produced by the turbine

Concept Introduction:

Write the change in molar enthalpy using the Peng-Robinson equation and residual properties.

${\underset{\_}{H}}_2-{\underset{\_}{H}}_1=\left({\underset{\_}{H}}_2-{\underset{\_}{H}}_2^{ig}\right)+\left({\underset{\_}{H}}_2^{ig}-{\underset{\_}{H}}_1^{ig}\right)-\left({\underset{\_}{H}}_1-{\underset{\_}{H}}_1^{ig}\right)$

Here, molar enthalpy at state 1 and 2 at inert gas is ${\underset{\_}{H}}_1^{ig}$ and ${\underset{\_}{H}}_2^{ig}$ respectively.

Write the reduced temperature $\left(T_r\right)$.

$T_r=\frac{T}{T_c}$

Here, critical temperature is $T_c$ and temperature is T.

Write the reduced pressure $\left(P_r\right)$.

$P_r=\frac{P}{P_c}$

Here, pressure is $P$.

Write the acentric factor in a manner analogous to Soave’s $m$.

$\kappa =0.37464+1.54226\omega -0.26993{\omega }^2$

Write the $\alpha$ as a function of the reduced temperature.

$\alpha ={\left[1+\kappa \left(1-T_r^{0.5}\right)\right]}^2$

Write the Peng-Robinson parameter a at the critical point.

$a_c=\frac{0.45724R^2{T}_c^2}{P_c}$

Write the van der Waals parameter $a$.

$a=a_c\alpha$

Write the van der Waals parameter $b$.

$b=\frac{0.07780R{T}_c}{P_c}$

Write the Peng-Robinson equation.

$P=\frac{RT}{\underset{\_}{V}-b}-\frac{a}{\underset{\_}{V}\left(\underset{\_}{V}+b\right)+b\left(\underset{\_}{V}-b\right)}$

Here, molar volume is $\underset{\_}{V}$, parameters of Robinson equation are a, b, gas constant is R, temperature and pressure is T and P respectively.

Write the compressibility factor.

$Z=\frac{P\underset{\_}{V}}{RT}$

Write the residual properties of A.

$A=\frac{aP}{R^2{T}^2}$

Write the residual properties of B.

$B=\frac{bP}{RT}$

Write the Peng Robinson Equation to calculation for residual molar internal energy.

$\begin{array}{l}\frac{{\underset{\_}{H}}^R}{R{T}_2}=\left\{\left(\frac{A}{B\sqrt{8}}\right)\left(1+\frac{\kappa \sqrt{T_r}}{\sqrt{a}}\right)\ln \left[\frac{Z+\left(1+\sqrt{2}\right)B}{Z+\left(1-\sqrt{2}\right)B}\right]\right\}\\ {\underset{\_}{H}}_2-{\underset{\_}{H}}_1=R{T}_2\left\{\left(\frac{A}{B\sqrt{8}}\right)\left(1+\frac{\kappa \sqrt{T_r}}{\sqrt{a}}\right)\ln \left[\frac{Z+\left(1+\sqrt{2}\right)B}{Z+\left(1-\sqrt{2}\right)B}\right]\right\}\end{array}$

Here, compressibility factor is Z, constants of residual properties are A and B.

(B)

Interpretation Introduction

Interpretation:

The rate of entropy generation

Concept Introduction:

Write the entropy balance equation for the turbine is adiabatic and steady state.

$0={\dot{m}}_{in}{\widehat{S}}_{in}-{\dot{m}}_{out}{\widehat{S}}_{out}+{\dot{S}}_{gen}$

Here, the rate of entropy generation is ${\dot{S}}_{gen}$, mass flow rate for inlet and outlet state is ${\dot{m}}_{in}$ and ${\dot{m}}_{out}$, specific entropy at inlet and outlet state is ${\widehat{S}}_{in}$ and ${\widehat{S}}_{out}$ respectively.

Write the change in molar entropy using residuals.

${\underset{\_}{S}}_2-{\underset{\_}{S}}_1=\left({\underset{\_}{S}}_2-{\underset{\_}{S}}_2^{ig}\right)+\left({\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}\right)-\left({\underset{\_}{S}}_1-{\underset{\_}{S}}_1^{ig}\right)$

Here, inert gas molar entropy for inlet and outlet streams is ${\underset{\_}{S}}_1^{ig}\text{and}{S}_2^{ig}$.

Write the ideal gas change in molar entropy.

${\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}=C_P^{*}\ln \left(\frac{T_2}{T_1}\right)-R\ln \left(\frac{P_2}{P_1}\right)$

Here, ideal gas heat capacity at constant pressure is $C_P^{*}$, gas constant is R, pressure and temperature at state 1 and 2 is $P_1,P_2,T_1,\text{and}{T}_2$ respectively.

Write the residual molar entropy for inlet stream calculated from Peng-Robinson equation.

$\begin{array}{l}\frac{{\underset{\_}{S}}^R}{R}=\ln \left(Z-B\right)-\left\{\left(\frac{A}{B\sqrt{8}}\right)\left(\frac{\kappa \sqrt{T_r}}{\sqrt{\alpha }}\right)\ln \left[\frac{Z+\left(1+\sqrt{2}\right)B}{Z+\left(1-\sqrt{2}\right)B}\right]\right\}\\ {\underset{\_}{S}}_1-{\underset{\_}{S}}_1^{ig}=R\left[\ln \left(Z-B\right)-\left\{\left(\frac{A}{B\sqrt{8}}\right)\left(\frac{\kappa \sqrt{T_r}}{\sqrt{\alpha }}\right)\ln \left[\frac{Z+\left(1+\sqrt{2}\right)B}{Z+\left(1-\sqrt{2}\right)B}\right]\right\}\right]\end{array}$

(C)

Interpretation Introduction

Interpretation:

The efficiency of the turbine

Concept Introduction:

Write the change in molar entropy using residuals for a reversible turbine.

$\begin{array}{l}{\underset{\_}{S}}_2-{\underset{\_}{S}}_1=\left({\underset{\_}{S}}_2-{\underset{\_}{S}}_2^{ig}\right)+\left({\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}\right)-\left({\underset{\_}{S}}_1-{\underset{\_}{S}}_1^{ig}\right)\\ \left({\underset{\_}{S}}_2-{\underset{\_}{S}}_2^{ig}\right)+\left({\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}\right)-\left({\underset{\_}{S}}_1-{\underset{\_}{S}}_1^{ig}\right)=0\end{array}$

Write the residual entropy of the VLE mixture.

${\underset{\_}{S}}_2-{\underset{\_}{S}}_2^{ig}=q{\underset{\_}{S}}^{R,V}+\left(1-q\right){\underset{\_}{S}}^{R,L}$ ]

Here, quality is q, residual molar entropy for liquid and vapor state is ${\underset{\_}{S}}^{R,L}$ and ${\underset{\_}{S}}^{R,V}$ respectively.